List Decoding Expander-Based Codes via Fast Approximation of Expanding CSPs: I

Abstract: We present near-linear time list decoding algorithms (in the block-length $n$) for expander-based code constructions. More precisely, we show that (i) For every $\delta \in (0,1)$ and $\epsilon > 0$, there is an explicit family of good Tanner LDPC codes of (design) distance $\delta$ that is $(\delta - \epsilon, O_\varepsilon(1))$ list decodable in time $\widetilde{\mathcal{O}}{\varepsilon}(n)$ with alphabet size $O\delta(1)$, (ii) For every $R \in (0,1)$ and $\epsilon > 0$, there is an explicit family of AEL codes of rate $R$, distance $1-R -\varepsilon$ that is $(1-R-\epsilon, O_\varepsilon(1))$ list decodable in time $\widetilde{\mathcal{O}}{\varepsilon}(n)$ with alphabet size $\text{exp}(\text{poly}(1/\epsilon))$, and (iii) For every $R \in (0,1)$ and $\epsilon > 0$, there is an explicit family of AEL codes of rate $R$, distance $1-R-\varepsilon$ that is $(1-R-\epsilon, O(1/\epsilon))$ list decodable in time $\widetilde{\mathcal{O}}{\varepsilon}(n)$ with alphabet size $\text{exp}(\text{exp}(\text{poly}(1/\epsilon)))$ using recent near-optimal list size bounds from [JMST25]. Our results are obtained by phrasing the decoding task as an agreement CSP [RWZ20,DHKNT19] on expander graphs and using the fast approximation algorithm for $q$-ary expanding CSPs from [Jer23], which is based on weak regularity decomposition [JST21,FK96]. Similarly to list decoding $q$-ary Ta-Shma's codes in [Jer23], we show that it suffices to enumerate over assignments that are constant in each part (of the constantly many) of the decomposition in order to recover all codewords in the list.